Percentages — typical problems with the formula and an explanation of every step

X% of a number

Percent, %

Number

formula: number × percent / 100

What percent is A of B

Number A

Number B

formula: A / B × 100%

Percent change (was → now)

Was

Now

formula: (now − was) / was × 100%

Original before discount / markup

Final number

Change, % (−15 is a discount)

formula: final / (1 + percent/100)

Add / subtract a percent

Number

Change, %

formula: number × (1 ± percent/100)

The number if X% of it equals A

Value A

That is percent, %

formula: A / percent × 100

📚 Theory: how to think about percentages

A percent is just one hundredth: 1% = 1/100 = 0.01. So "15% of 240" is 240 × 0.15 = 36. Every percent problem reduces to multiplying or dividing by such a fraction.

A common trap: a discount cannot be "undone" by the same percent. If a price went up 20% and then down 20%, you do not get back to the original: 100 → 120 → 96. That is why "the number before the discount" is computed by division, not by subtracting the percent.

Percent change is always measured from the original value: growth from 80 to 100 is +25% (20/80), while a drop from 100 to 80 is −20% (20/100). The same difference — different percentages!

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